Botox Optimization Algorithm: A New Human-Based Metaheuristic Algorithm for Solving Optimization Problems

This paper introduces the Botox Optimization Algorithm (BOA), a novel metaheuristic inspired by the Botox operation mechanism. The algorithm is designed to address optimization problems, utilizing a human-based approach. Taking cues from Botox procedures, where defects are targeted and treated to enhance beauty, the BOA is formulated and mathematically modeled. Evaluation on the CEC 2017 test suite showcases the BOA’s ability to balance exploration and exploitation, delivering competitive solutions. Comparative analysis against twelve well-known metaheuristic algorithms demonstrates the BOA’s superior performance across various benchmark functions, with statistically significant advantages. Moreover, application to constrained optimization problems from the CEC 2011 test suite highlights the BOA’s effectiveness in real-world optimization tasks.


Introduction
Optimization problems, characterized by multiple feasible solutions, involve finding the best solution among them.Mathematically, these problems consist of decision variables, constraints, and an objective function.The optimization process aims to determine optimal values for decision variables, adhering to constraints while optimizing the objective function.Numerous real-world applications in science, engineering, industry, and technology necessitate effective optimization techniques.Two main approaches, deterministic and stochastic, address these challenges.Deterministic approaches, including gradient-based and non-gradient-based methods, excel in handling simpler problems but face limitations in complexity and local optima traps.To address complex, nonlinear, and high-dimensional challenges, researchers have developed stochastic approaches, acknowledging the limitations of deterministic methods in practical optimization scenarios [1][2][3][4][5][6].
Metaheuristic algorithms represent a widely employed stochastic approach for effective optimization problem-solving.Leveraging random search, random operators, and trial-and-error processes, these algorithms yield suitable solutions.The optimization process initiates with the random generation of candidate solutions, progressively enhancing them through iterations.The final output is the best-improved candidate solution.While the inherent randomness poses challenges in guaranteeing a global optimal solution, solutions obtained from metaheuristic algorithms are considered to be quasi-optimal due to their proximity to the global optimum.The pursuit of more effective quasi-optimal solutions, closely aligning with the global optimum, drives the development of various metaheuristic algorithms [7,8].
For metaheuristic algorithms to effectively address optimization problems, they must conduct thorough searches at both the global and local levels within the problem-solving space.Global search, aligned with exploration, denotes the algorithm's proficiency in extensively exploring the problem-solving space to identify the region containing the primary optimum and avoid local optima.Local search, associated with exploitation,

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Introducing the BOA involves emulating the Botox injection process, drawing inspiration from enhancing facial beauty by addressing defects in specific facial areas.

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BOA theory is described and then mathematically modeled.• The BOA's performance is rigorously assessed using the CEC 2017 test suite, showcas- ing its efficacy in solving optimization problems.• The algorithm's robustness is further tested in handling real-world applications, particularly in optimizing twenty-two constrained problems from the CEC 2011 test suite.
• The BOA's performance is objectively compared with twelve established metaheuristic algorithms, establishing its competitive edge and effectiveness.
This paper follows a structured outline: Section 2 encompasses a comprehensive literature review.Section 3 introduces and models the Botox Optimization Algorithm.Section 4 presents simulation studies and results.The efficacy of the BOA in real-world applications is explored in Section 5.The paper concludes with Section 6, offering conclusions and suggestions for future research.

Literature Review
Metaheuristic algorithms draw inspiration from diverse sources, such as natural phenomena, living organisms' lifestyles, the laws of physics, biology, human interactions, and game rules.Classified into five groups based on their design principles, these are swarmbased, evolutionary-based, physics-based, human-based, and game-based approaches.
Evolutionary-based metaheuristic algorithms derive inspiration from the biological sciences, genetics, survival of the fittest, natural selection, and random operators.Prominent algorithms in this group include the Genetic Algorithm (GA) [31] and Differential Evolution (DE) [32], designed to emulate reproduction and Darwin's theory of evolution, and to incorporate random operators like mutation, crossover, and selection.Artificial Immune Systems (AISs) are modeled after the human body's defense system [33].Other algorithms in this category encompass Genetic Programming (GP) [34], Cultural Algorithm (CA) [35], and Evolution Strategy (ES) [36].
Game-based metaheuristic algorithms are formulated by simulating player behavior, influential figures, and the rules of various individual and team games.Algorithms like Football Game-Based Optimization (FGBO) [63] and Volleyball Premier League (VPL) [64] are inspired by modeling league matches.The Hide Object Game Optimizer (HOGO) [65] is designed based on players' attempts to locate hidden objects on the playing field.The Darts Game Optimizer (DGO) [66] incorporates the skill of players throwing darts to earn more points.The Orientation Search Algorithm (OSA) [67] emulates players' movements directed by referees.Other game-based metaheuristic algorithms include the Dice Game Optimizer (DGO) [68], Golf Optimization Algorithm (GOA), League Championship Algorithm (LCA) [6], Ring Toss Game-Based Optimization (RTGBO) [69], and Puzzle Optimization Algorithm (POA) [70].
In addition to the original versions of metaheuristic algorithms, many researchers have tried to improve the performance of existing algorithms by developing their improved versions, such as the Enhanced Snake Optimizer (ESO) [71], Improved Sparrow Search Algorithm (ISSA) [72], and multi-strategy-based Adaptive Sine-Cosine Algorithm (ASCA) [73].
To the best of our knowledge, as gleaned from the literature review, no metaheuristic algorithm inspired by the human activity of Botox injections has been introduced thus far.The process of enhancing facial beauty by injecting substances to eliminate facial defects presents an intelligent methodology that could serve as the foundation for a novel metaheuristic algorithm.To bridge this research gap in metaheuristic algorithm studies, this paper introduces a new human-based metaheuristic algorithm, grounded in the mathematical modeling of Botox injections in specific facial areas, as elaborated in the subsequent section.

Botox Optimization Algorithm
Within this section, the Botox Optimization Algorithm (BOA) is elucidated, beginning with an exploration of its theory and source of inspiration.Following this, the mathematical modeling of the implementation steps for the proposed BOA approach is detailed.

Inspiration of BOA
Enhancing facial beauty is a significant and intricate concern for many individuals, with the emergence of facial wrinkles often causing distress.Wrinkles result from the repetitive contraction of underlying facial muscles and dermal atrophy.To address this issue, small doses of botulinum toxin are strategically injected into specific overactive muscles.This injection induces localized muscle relaxation, subsequently leading to the smoothing of the skin in these hyperactive muscle areas [74].Botulinum toxin, a potent neurotoxin protein derived from the bacterium Clostridium botulinum, is employed for this purpose.The administration of this toxin results in the targeted muscles being temporarily paralyzed, preventing the formation of wrinkles in the treated area [75].Botox, the cosmetic use of botulinum toxin, gained approval from the U.S. Food and Drug Administration (FDA) in 2002 for treating glabellar complex muscles responsible for frown lines, and in 2013 for addressing lateral orbicularis oculi muscles associated with crow's feet [76].
Botox exerts a significant impact on diminishing facial wrinkles and enhancing facial aesthetics.The strategic injection of Botox into specific facial areas to eliminate wrinkles serves as an intelligent process, forming the foundational concept behind the design of the approach proposed by the BOA.

Algorithm Initialization
The proposed BOA methodology operates as a population-based optimizer, leveraging the collective search capabilities of its participants in an iterative process to generate viable solutions for optimization problems.In this context, individuals seeking Botox injections constitute the BOA population.Each member contributes to decision variable values based on their position in the problem-solving space, mathematically represented as a vector.This vector, encapsulating decision variables, forms the population matrix outlined in Equation (1); initialization of each BOA member's position is achieved through random assignment using Equation (2): where X is the BOA population matrix, → X i is the ith BOA member (candidate solution), x i,d is its dth dimension in the search space (decision variable), N is the number of population members, m is the number of decision variables, r i,d are random numbers from interval [0, 1], and lb d and ub d are the lower bound and upper bound of the dth decision variable, respectively.
Given that each member in the BOA population represents a candidate solution for the problem, the associated objective function of the problem can be assessed for each individual.Consequently, the array of objective function values can be depicted as a vector, as per Equation (3): where → F is the vector of the evaluated objective function and F i is the evaluated objective function based on the ith BOA member.
The assessed objective function values serve as reliable criteria for appraising the quality of candidate solutions.Consequently, the optimal member of the BOA corresponds to the best value achieved for the objective function, while the suboptimal member aligns with the worst value.Given that the position of BOA population members and their objective function values are updated in each iteration, the best candidate solution undergoes regular updates.

Mathematical Modeling of BOA
The BOA approach, a population-based optimizer, adeptly furnishes viable solutions for optimization problems through an iterative process.In the BOA's design, inspiration is drawn from the Botox injection mechanism to update the position of population members within the search space.The schematic of Botox injection and its simulation to design the proposed BOA approach is shown in Figure 1.
Each individual seeking Botox injections represents a member of the BOA population.The BOA design mirrors the process of a doctor injecting Botox into specific facial muscles to diminish wrinkles and enhance beauty.Similarly, in the BOA approach, improvement to a candidate solution involves adding a designated value, akin to Botox, to select decision variables.
In the design of the BOA, it is considered that the number of facial muscles that need to be injected with Botox decreases during the iterations of the algorithm.Therefore, the number of selected muscles (i.e., decision variables) for Botox injection is determined by using Equation (4): where N b is the number of muscles requiring Botox injection and t is the current value of the iteration counter.
When the applicant visits the doctor, the doctor decides which muscles to inject Botox into, based on the person's face and wrinkles.Inspired by this fact, in BOA design, the variables to be injected are selected for each population member using Equation (5).It should be noted that the muscles that are chosen for Botox injection should not be repeated, which is considered in Equation (5): (5) Thus, CBS i is the set of candidate decision variables of the ith population member that are selected for Botox injection, and d j is the position of the jth decision variable selected for Botox injection.(5) Thus,  is the set of candidate decision variables of the th population member that are selected for Botox injection, and  is the position of the th decision variable selected for Botox injection.In the BOA design, akin to the doctor's discretion in determining the drug quantity for Botox injection based on expertise and patient needs, the amount of Botox injection for each population member is computed using Equation ( 6): where  ⃗ = ( , , … ,  , , … ,  , ) is the considered amount for Botox injection to the th member,  ⃗ is the mean population position (i.e.,  ⃗ = ∑  ⃗ ),  is the total number of iterations, and  ⃗ is the best population member.After Botox injection into the facial muscles, the appearance of the face changes, with the disappearance of wrinkles.In the BOA design, based on the simulation of Botox injection to the facial muscles, first, a new position is calculated for each BOA member based on Botox injection using Equation (7); then, if the value of the objective function is improved, this new position replaces the previous position of the corresponding member according to Equation ( 8  In the BOA design, akin to the doctor's discretion in determining the drug quantity for Botox injection based on expertise and patient needs, the amount of Botox injection for each population member is computed using Equation (6): where ) is the considered amount for Botox injection to the ith member, → X mean is the mean population position (i.e., T is the total number of iterations, and → X best is the best population member.After Botox injection into the facial muscles, the appearance of the face changes, with the disappearance of wrinkles.In the BOA design, based on the simulation of Botox injection to the facial muscles, first, a new position is calculated for each BOA member based on Botox injection using Equation (7); then, if the value of the objective function is improved, this new position replaces the previous position of the corresponding member according to Equation (8): After updating the position of all BOA members in the search space, the first iteration of the algorithm is completed.Then, based on the updated values, the algorithm enters the next iteration, and the process of updating the BOA population members continues until the last iteration, based on Equations ( 4)- (8).In each iteration, the best obtained candidate solution → X best is also updated and saved.After the full implementation of the proposed BOA approach, the best candidate solution → X best stored during the iterations of the algorithm is introduced as the solution to the given problem.The steps of BOA implementation are presented in the form of a flowchart in Figure 2, and its pseudocode is shown in Algorithm 1. Update number of decision variables for Botox injections using Equation (4).8.
For i = 1 to N 9.
Determine the variables that are considered for Botox injection using Equation (5).10.
Calculate the amount of Botox injection using Equation (6).11.
For j = 1 to N b 12.
Calculate the new position of the ith BOA member using Equation (7)

Computational Complexity of the BOA
In this subsection, the computational complexity of the BOA is evaluated.The preparation and initialization steps of the BOA for an optimization problem have a computational complexity equal to O(Nm), where N is the number of population members and m is the number of decision variables of the problem.In each iteration, the position of the population members is updated and the corresponding objective function is also evaluated.Therefore, the BOA update process has a computational complexity equal to O(NmT), where T is the maximum number of iterations of the algorithm.According to this, the total computational complexity of the proposed BOA approach is equal to O(Nm(1 + T)).

Computational Complexity of the BOA
In this subsection, the computational complexity of the BOA is evaluated.The preparation and initialization steps of the BOA for an optimization problem have a computational complexity equal to (), where  is the number of population members and

Population Diversity, Exploration, and Exploitation Analysis
The population diversity of the BOA refers to the distribution of population members within the problem space, which plays a critical role in monitoring the search processes of the algorithm.Essentially, this metric indicates whether the population members are focused on exploration or exploitation.By measuring the diversity of the BOA population, it becomes possible to gauge and adapt the algorithm's capacity to explore and exploit a collective group effectively.Various definitions of diversity have been put forth by researchers.Pant [77] defined diversity according to Equations ( 9) and (10): where N is the number of population members, m is the number of problem dimensions, and x d is the mean of the entire population in the dth dimension.Hence, the percentage of exploration and exploitation of the population for each iteration can be defined by Equations ( 11) and (12), respectively: In this subsection, the analysis of population diversity, exploration, and exploitation is evaluated on twenty-three standard benchmark functions, consisting of 7 unimodal functions (F1 to F7) and 16 multimodal functions (F8 to F23).A full description of these benchmark functions is available in [78].
Figure 3 illustrates the exploration-exploitation ratio of the BOA method throughout the iteration process, offering visual support for analyzing how the algorithm balances global and local search strategies.Also, the results of the analysis of population diversity, exploration, and exploitation are reported in Table 1.The simulation results show that the BOA has favorable population diversity, where it has high values in the first iteration, while the values of this index are low in the last iteration.Also, based on the obtained results, in most cases the exploration-exploitation ratio of the BOA is close to 0.00%:100%.The findings obtained from this analysis confirm that the proposed BOA approach, by creating the appropriate population diversity during the iterations of the algorithm, provides a favorable performance in managing exploration and exploitation, and in balancing them during the search process.

Simulation Studies and Results
In this section, the performance of the proposed BOA approach in handling optimization tasks is evaluated.

Performance Comparison
To assess the BOA's effectiveness in addressing optimization problems, its results were juxtaposed with those of twelve prominent metaheuristic algorithms: the GA [31], PSO [11], GSA [40], TLBO [51], MVO [42], GWO [22], WOA [26], MPA [25], TSA [17], RSA [28], AVOA [23], and WSO [21].These twelve algorithms were selected from the numerous algorithms available in the literature.The reasons for choosing these twelve algorithms were as follows: the GA and PSO are among the first and most famous metaheuristic algorithms.The GSA, TLBO, MVO, GWO, and WOA are among the most cited metaheuristic algorithms that have been used in various optimization applications.The MPA, TSA, RSA, AVOA, and WSO approaches are among the recently published successful metaheuristic algorithms that have attracted the attention of many researchers in this short period of time.Comparing the proposed BOA approach with these twelve selected metaheuristic algorithms is a valuable comparison, after which the efficiency of the BOA will have been tested well.Table 2 outlines the control parameter values for the competing algorithms.The evaluation of the simulation results incorporates six statistical metrics: mean, best, worst, standard deviation (std), median, and rank.The mean index values were utilized for ranking the metaheuristic algorithms concerning each benchmark function.

Evaluation of the CEC 2017 Test Suite
In this section, the performance of the BOA and competing algorithms is evaluated using the CEC 2017 test suite, considering problem dimensions (number of decision variables) equal to 10, 30, 50, and 100.The CEC 2017 test suite comprises thirty benchmark functions, including three unimodal functions (C17-F1 to C17-F3), seven multimodal functions (C17-F4 to C17-F10), ten hybrid functions (C17-F11 to C17-F20), and ten composition functions (C17-F21 to C17-F30).The C17-F2 function is excluded due to its unstable behavior, as described in [79].
The results of employing the BOA approach and competing algorithms on the CEC 2017 test suite are presented in Table 3. Boxplot diagrams depicting the performance of the BOA and competing algorithms in optimizing the CEC 2017 test suite are illustrated in Figure 4.The outcomes indicate that the BOA outperformed other optimizers, ranking as the top performer for functions C17-F1, C17-F3 to C17-F21, C17-F23, C17-F24, and C17-F27 to C17-F30.F27 to C17-F30.
Overall, the BOA demonstrated its efficacy in providing effective solutions for the CEC 2017 test suite, showcasing a commendable ability to explore, exploit, and maintain balance throughout the search process.The simulation results establish the BOA's superior performance over competing algorithms, securing the top rank as the best optimizer for handling the CEC 2017 test suite.Overall, the BOA demonstrated its efficacy in providing effective solutions for the CEC 2017 test suite, showcasing a commendable ability to explore, exploit, and maintain balance throughout the search process.The simulation results establish the BOA's superior performance over competing algorithms, securing the top rank as the best optimizer for handling the CEC 2017 test suite.

Statistical Analysis
In this section, a statistical analysis was performed on the performances of the BOA and rival algorithms to assess the significance of the BOA's superiority from a statistical perspective.The Wilcoxon signed-rank test [80], a non-parametric test for matched or paired data, was employed for this purpose.This test helps determine whether there is a significant difference between the averages of two data samples.The results of the Wilcoxon signed-rank test, presented in Table 4, indicate instances where the BOA exhibits statistically significant superiority over the respective competing algorithms, with a p-value criterion of less than 0.05.

Discussion
In this subsection, the performance of the BOA compared to competing algorithms is discussed.The CEC 2017 test suite has different types of objective functions.
Unimodal functions C17-F1 and C17-F3 have only one main optimum (i.e., global optimum), and for that reason they are suitable criteria for measuring the exploitation ability of metaheuristic algorithms.Analysis of the simulation results shows that the proposed BOA approach, with a strong performance in local search, has superior performance against all twelve competing algorithms for handling unimodal functions.Therefore, as the first strength, the superiority of the BOA in exploitation is confirmed against competing algorithms.Multimodal functions C17-F4 to C17-F10, in addition to the main optimum (i.e., the global optimum), also have a number of local optima, which challenge the exploration ability of metaheuristic algorithms.The findings obtained from the simulation results show that the BOA, with global search management, was able to achieve the rank of the best optimizer in the competition with the compared algorithms to handle the functions C17-F4 to C17-F10.The simulation results confirm that, as the second strength, the BOA has a better exploration ability to manage global search compared to competing algorithms.
Hybrid functions C17-F11 to C17-F20 and composition functions C17-F21 to C17-F30 are complex optimization problems that challenge the performance of metaheuristic algorithms in establishing a balance between exploration and exploitation.The simulation results of these functions show that the BOA was able to achieve the rank of the best optimizer in most of these benchmark functions, except for C17-F22, C17-F25, and C17-F26.The simulation results confirm that the BOA is highly capable of balancing exploration and exploitation when facing complex optimization problems.Therefore, as a third strength, the superiority of the BOA in balancing exploration and exploitation is confirmed compared to competing algorithms.
In addition, the statistical analysis of the Wilcoxon signed-rank test and the values obtained for the p-value index, as the fourth strength, confirm that the BOA has a significant statistical superiority compared to all twelve competing algorithms.

BOA for Real-World Applications
In this section, the effectiveness of the proposed BOA approach in addressing real-world optimization tasks is evaluated.To this end, twenty-two constrained optimization problems from the CEC 2011 test suite, along with four engineering design problems, are utilized.

Evaluation of CEC 2011 Test Suite
In this subsection, the performance of the BOA in optimizing the CEC 2011 test suite, which comprises twenty-two constrained optimization problems from real-world applications, is assessed.Detailed descriptions and information about the CEC 2011 test suite can be found in [81].The results of employing the BOA and competing algorithms on the CEC 2011 test suite are presented in Table 5, and the boxplot diagrams illustrating the performance of the BOA and competing algorithms are depicted in Figure 5.The optimization outcomes highlight that the BOA effectively generated suitable solutions for this test suite, showcasing a balanced exploration and exploitation throughout the search process.Notably, the BOA emerges as the top optimizer for solving functions C11-F1 to C11-F22, demonstrating superior performance in comparison to competing algorithms.Statistical analysis, specifically the Wilcoxon signed-rank test, further validates the significant statistical superiority of the BOA in these evaluations.

Pressure Vessel Design Problem
The design of the pressure vessel in engineering aims primarily to minimize construction costs, as illustrated in Figure 6.The mathematical representation of pressure vessel design is defined as follows [82]: Consider

Pressure Vessel Design Problem
The design of the pressure vessel in engineering aims primarily to minimize construction costs, as illustrated in Figure 6.The mathematical representation of pressure vessel design is defined as follows [82]   Consider: The outcomes derived from applying the BOA and rival algorithms to optimize pressure vessel design are documented in Tables 6 and 7.According to the results, the BOA yielded the optimal solution for this design, with design variable values of (0.7781685, 0.3846492, 40.319615, 200) and an objective function value of 5885.3263.The convergence curve of the BOA throughout the discovery of the optimal solution for pressure vessel design is depicted in Figure 7. Examination of the optimization results indicates that the BOA exhibits superior performance in addressing pressure vessel design challenges, outperforming competing algorithms.

Speed Reducer Design Problem
The design of a speed reducer is a practical engineering application focused on minimizing the weight of the speed reducer, as illustrated in Figure 8.The mathematical model for the design of the speed reducer is outlined in [83,84]:    Consider: X = x 1, x 2 , x 3 , x 4 , x 5 , x 6 , x 7 = b, m, p, l ).Subject to x 2 x 3 ã 2 + 16.9 × 10 6 − 1 ≤ 0, The outcomes of implementing the BOA and competing optimizers to address the speed reducer design challenges are documented in Tables 8 and 9.The BOA yielded the optimal solution for this design, characterized by design variable values (3.5, 0.7, 17, 7.3, 7.8, 3.3502147, 5.2866832) and an objective function value of 2996.3482.The convergence curve, depicting the BOA's performance in optimizing the speed reducer design, is illustrated in Figure 9.The analysis of the simulation results confirms that the BOA demonstrated more effective performance in tackling the speed reducer design compared to its competitors.

Welded Beam Design
The design of a welded beam poses a real-world engineering challenge, intending to minimize the fabrication cost of the beam, as depicted in Figure 10.The mathematical model governing the welded beam design is outlined as follows [26]

Welded Beam Design
The design of a welded beam poses a real-world engineering challenge, intending to minimize the fabrication cost of the beam, as depicted in Figure 10.The mathematical model governing the welded beam design is outlined as follows [26]: Biomimetics 2024, 9, x FOR PEER REVIEW 28 of 34 Figure 11.The simulation results underscore the effectiveness of the BOA in addressing the welded beam design problem, showcasing superior performance compared to competing algorithms.Consider: The optimization outcomes for the welded beam design, utilizing the BOA and competing algorithms, are outlined in Tables 10 and 11.The BOA yielded the optimal solution for this design, with design variable values set at (0.2057296, 3.4704887, 9.0366239, 0.2057296), resulting in an objective function value of 1.7246798.The convergence process of the BOA towards the optimal solution for the welded beam design is illustrated in Figure 11.The simulation results underscore the effectiveness of the BOA in addressing the welded beam design problem, showcasing superior performance compared to competing algorithms.

Tension/Compression Spring Design Problem
The engineering challenge in tension/compression spring design is to minimize the weight of the spring, as depicted in Figure 12.The mathematical model for tension/compression spring design is outlined as follows [26]: Consider The optimization outcomes for tension/compression spring design using the BOA and competing algorithms are outlined in Tables 12 and 13.The BOA yielded the optimal solution for this design, with design variable values of (0.0516891, 0.3567177, 11.288966) and an objective function value of 0.0126019.The convergence curve depicting the BOA's performance in optimizing the tension/compression spring design is illustrated in Figure 13.The simulation results demonstrate that the BOA exhibited superior performance compared to competing algorithms by delivering improved outcomes for tension/compression spring design.

Tension/Compression Spring Design Problem
The engineering challenge in tension/compression spring design is to minimize the weight of the spring, as depicted in Figure 12.The mathematical model for tension/compression spring design is outlined as follows [26]:     The optimization outcomes for tension/compression spring design using the BOA and competing algorithms are outlined in Tables 12 and 13.The BOA yielded the optimal solution for this design, with design variable values of (0.0516891, 0.3567177, 11.288966) and an objective function value of 0.0126019.The convergence curve depicting the BOA's performance in optimizing the tension/compression spring design is illustrated in Figure 13.
The simulation results demonstrate that the BOA exhibited superior performance compared to competing algorithms by delivering improved outcomes for tension/compression spring design.

Conclusions and Future Works
In this paper, motivated by the No Free Lunch (NFL) theorem, a new human-based metaheuristic algorithm called the Botox Optimization Algorithm (BOA) was introduced, mimicking the human action of Botox injections.The originality of the proposed BOA approach was confirmed based on the best knowledge obtained from the literature review, where no metaheuristic algorithm based on Botox injection modeling has been designed so far.The fundamental inspiration of the BOA is the injection of Botox into the areas of the face in order to remove defects from the face and increase facial beauty.The theory of the BOA was stated, and the various stages of its implementation were mathematically modeled based on the simulation of Botox injection.The performance of the BOA was evaluated on the CEC 2017 test suite.The optimization results showed that the BOA has a high ability to balance exploration and exploitation during the search process.To measure the quality of the BOA, the obtained results were compared with the performance of twelve well-known metaheuristic algorithms.The simulation results showed that the BOA outperformed competing algorithms by providing better results in most benchmark functions.Using statistical analysis, it was shown that the BOA has significant statistical superiority over competing algorithms.Also, the implementation of the BOA on twentytwo constrained optimization problems from the CEC 2011 test suite showed the ability of the proposed approach to handle real-world applications.
After introducing the proposed BOA approach, several research paths can be considered for further studies:

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Binary BOA: The real version of the BOA is detailed and explained thoroughly in this paper.Nonetheless, many scientific optimization issues, like feature selection, require the use of binary versions of metaheuristic algorithms for efficient optimization.
Consequently, developing the binary version of the BOA (BBOA) is a notable focus of this research.

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Multi-objective BOA: Optimization problems are classified based on the number of objective functions, which are either single-objective or multi-objective.To find an optimal solution, many problems require the consideration of multiple objective functions simultaneously.Hence, exploring the potential of developing a multi-objective version of the BOA (MOBOA) to address multi-objective optimization dilemmas is another area of research highlighted in this paper.

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Hybrid BOA: Researchers have always been intrigued by the idea of merging multiple metaheuristic algorithms to leverage the strengths of each and establish a more efficient hybrid strategy.Hence, a potential future research endeavor includes crafting hybrid versions of the BOA.• Tackle new domains: Exploring opportunities for employing the BOA in tackling practical applications and optimizing problems within various scientific fields, like robotics, renewable energy, chemical engineering, and image processing, is a focus for future research proposals.

Figure 1 .
Figure 1.Schematic diagram of the Botox injection and the proposed BOA.

Figure 1 .
Figure 1.Schematic diagram of the Botox injection and the proposed BOA.

Figure 3 .
Figure 3. Exploration and exploitation of the BOA.

Figure 4 .
Figure 4. Boxplot representations illustrating the performances of the BOA and rival algorithms on the CEC 2017 test suite.

Figure 4 .
Figure 4. Boxplot representations illustrating the performances of the BOA and rival algorithms on the CEC 2017 test suite.

Figure 5 .
Figure 5. Boxplot diagrams of the BOA and competing algorithms' performances on the CEC 2011 test suite.

:Figure 5 .
Figure 5. Boxplot diagrams of the BOA and competing algorithms' performances on the CEC 2011 test suite.

Figure 6 .
Figure 6.Schematic of pressure vessel design.The thickness of the shell is  , the thickness of the head is  , the length of cylindrical shell is , and the inner radius is .

Figure 6 .
Figure 6.Schematic of pressure vessel design.The thickness of the shell is T s , the thickness of the head is T h , the length of cylindrical shell is L, and the inner radius is R.

Figure 6 .
Figure 6.Schematic of pressure vessel design.The thickness of the shell is  , the thickness of the head is  , the length of cylindrical shell is , and the inner radius is .

Figure 7 .
Figure 7.The BOA's performance convergence curve on pressure vessel design.

Figure 8 .
Figure 8. Schematic of speed reducer design.The face width is , the number of teeth on the pinion is , the module of teeth is , the length of the second shaft between bearings is  , the length of the first shaft between bearings is  , the second shaft's diameter is  , and the first shaft's diameter is  .

Figure 8 .
Figure8.Schematic of speed reducer design.The face width is b, the number of teeth on the pinion is z, the module of teeth is m, the length of the second shaft between bearings is l 2 , the length of the first shaft between bearings is l 1 , the second shaft's diameter is d 2 , and the first shaft's diameter is d 1 .

Figure 9 .
Figure 9.The BOA's performance convergence curve on speed reducer design.

Figure 9 .
Figure 9.The BOA's performance convergence curve on speed reducer design.

Figure 10 .
Figure 10.Schematic of welded beam design.The bar height is t, the weld thickness is h, the thickness of bar is b, and the length of clamped bar is l.

Figure 10 .
Figure 10.Schematic of welded beam design.The bar height is , the weld thickness is ℎ, the thickness of bar is , and the length of clamped bar is .

Figure 11 .
Figure 11.The BOA's performance convergence curve on welded beam design.

Figure 11 .
Figure 11.The BOA's performance convergence curve on welded beam design.

Figure 12 .
Figure 12.Schematic of tension/compression spring design.The wire's diameter is , the number of active coils is , and the mean coil's diameter is .

Figure 13 .
Figure 13.The BOA's performance convergence curve on tension/compression spring design.6.Conclusions and Future WorksIn this paper, motivated by the No Free Lunch (NFL) theorem, a new human-based metaheuristic algorithm called the Botox Optimization Algorithm (BOA) was introduced, mimicking the human action of Botox injections.The originality of the proposed BOA approach was confirmed based on the best knowledge obtained from the literature review, where no metaheuristic algorithm based on Botox injection modeling has been designed

Figure 12 .
Figure 12.Schematic of tension/compression spring design.The wire's diameter is , the number of active coils is , and the mean coil's diameter is .

Figure 13 .
Figure 13.The BOA's performance convergence curve on tension/compression spring design.6.Conclusions and Future WorksIn this paper, motivated by the No Free Lunch (NFL) theorem, a new human-based metaheuristic algorithm called the Botox Optimization Algorithm (BOA) was introduced, mimicking the human action of Botox injections.The originality of the proposed BOA approach was confirmed based on the best knowledge obtained from the literature review, where no metaheuristic algorithm based on Botox injection modeling has been designed so far.The fundamental inspiration of the BOA is the injection of Botox into the areas of

Figure 13 .
Figure 13.The BOA's performance convergence curve on tension/compression spring design.

Table 1 .
Population diversity, exploration, and exploitation percentage results.

Table 3 .
Optimization results of the CEC 2017 test suite.

Table 3 .
Optimization results of the CEC 2017 test suite.

Table 5 .
Optimization results of the CEC 2011 test suite.

Table 6 .
Performance of optimization algorithms on the pressure vessel design problem.

Table 7 .
Statistical results of optimization algorithms on the pressure vessel design problem.

Table 9 .
Statistical results of optimization algorithms on the speed reducer design problem.

Table 8 .
Performance of optimization algorithms on the speed reducer design problem.

Table 9 .
Statistical results of optimization algorithms on the speed reducer design problem.

Table 10 .
Performance of optimization algorithms on the welded beam design problem.

Table 11 .
Statistical results of optimization algorithms on the welded beam design problem.

Table 10 .
Performance of optimization algorithms on the welded beam design problem.

Table 11 .
Statistical results of optimization algorithms on the welded beam design problem.

Table 12 .
Performance of optimization algorithms on the tension/compression spring design problem.

Table 13 .
Statistical results of optimization algorithms on the tension/compression spring design problem.

Table 12 .
Performance of optimization algorithms on the tension/compression spring design problem.

Table 13 .
Statistical results of optimization algorithms on the tension/compression spring design problem.

Table 13 .
Statistical results of optimization algorithms on the tension/compression spring design problem.